This presentation is just for knowledge

2. MECHANICS-1

3. PRESENTED BY:
MISBAH FAZAL(Physics 1 sem)

4. PRESENTATION TOPIC:
 Determine Radius Of Gyration
About an Axis Through the Center
Gravity for Compound Pendulum

5. AIM:
To determine radius of gyration about
an axis through the center of gravity for
the compound pendulum.
Apparatus and Accessories:
(i) A bar pendulum.
(ii) a knife–edge with a platform.
(iii) a sprit level.
(iv) a precision stop watch.
(v) a meter scale and.
(vi) a telescope.

6. THEORY:
A body of irregular shape is pivoted
about a horizontal frictionless axis through P and
is
displaced from its equilibrium position by an angle
θ. In the equilibrium position the center of
gravity G of the body is vertically below P. The
distance GP is l and the mass of the body is m.
The restoring torque for an angular displacement θ
is

7. DIAGRAM:

8. where I is the moment of inertia of the
body through the axis P.
Eq. (2) represents a simple harmonic
motion and hence the time
where IG is the moment of inertia of the
body about an axis parallel with axis of oscillation
and passing
through the center of gravity G.
IG = mK2
…(4)
where K is the radius of gyration about the axis
passing through G.

9. The time period of a simple pendulum of length
L, is given by
Comparing with Eq. (5) we get

10. This is the length of “equivalent simple pendulum”. If
all the mass of the body were concentrated
at a point O (See Fig.1) such that , we would have a
simple pendulum with the same time period. The point
O is called the ‘Centre of Oscillation’. Now from Eq. (7
quadratic equation in l.
Equation 6 has two roots l1 and l2 such that

11. Thus both
L1 and
L2 are positive. This means that on one side of C.G there
are two positions of
the centre of suspension about which the time periods
are the same. Similarly, there will be a pair
of positions of the centre of suspension on the other side
of the C.G about which the time periods
will be the same. Thus there are four positions of the
centers of suspension, two on either side of
the C.G, about which the time periods of the pendulum
would be the same. The distance between
two such positions of the centers of suspension,
asymmetrically located on either side of C.G, is

12. the length L of the simple equivalent pendulum.
Thus, if the body was supported on a parallel
axis through the point O (see Fig. 1), it would
oscillate with the same time period T as when
supported at P. Now it is evident that on either
side of G, there are infinite numbers of such pair
of points satisfying Eq. (9). If the body is
supported by an axis through G, the time period
of
oscillation would be infinite. From any other axis
in the body the time period is given by Eq. (5).

13. By determining L, L1 and
L2 graphically for a particular value of T, the
acceleration due to gravity g at that place and the
radius of gyration K of the compound pendulum
can be determined
Description:
The bar pendulum consists of a metallic bar of about
one meter long. A series of circular holes
each of approximately 5 mm in diameter are made
along the length of the bar

14. The bar is suspended from a horizontal knife-
edge passing through any of the holes The knife-
edge, in turn,is fixed in a platform provided
without the screws. By adjusting the rear screw
the platform can be made horizontal.
Graph
between time
and distance.

15. Procedure:
(i) Suspend the bar using the knife edge of the hook
through a hole nearest to one end of the bar.
With the bar at rest, focus a telescope so that the
vertical cross-wire of the telescope is coincident
with the vertical mark on the bar.
(ii) Allow the bar to oscillate in a vertical plane with
small amplitude (within 40of arc).
(iii) Note the time for 20 oscillations by a precision
stop-watch by observing the transits of the vertical
line on the bar through the telescope. Make this
observation three times and find the mean time t for
20 oscillations. Determine the time period T.

16. (iv) Measure the distance d of the axis of the
suspension, i.e. the hole from one of the edges of
the bar by a meter scale.
(v) Repeat operation (i) to (iv) for the other holes till
C.G of the bar is approached where the time
period becomes very large.
(vi) Invert the bar and repeat operations (i) to (v) for
each hole starting from the extreme top.
(vii) Draw a graph with the distance d of the holes as
abscissa and the time period T as ordinate.
The nature of graph will be as shown in Fig. 3.

17. Draw the horizontal line ABCDE parallel to the X-axis.
Here A, B, D and E represent the point
of intersections of the line with the curves. Note that
the curves are symmetrical about a vertical
line which meets the X-axis at the point G, which gives
the position of the C.G of the bar. This vertical line
intersects with the line ABCDE at C. Determine the
length AD and BE and find the length L of the
equivalent simple pendulum from

18. Find also the time period T corresponding to the line
ABCDE and then compute the value of g. Draw several
horizontal lines parallel to X-axis and adopting the
above procedure find the value of g for each horizontal
line. Calculate the mean value of g. Alternatively, for
each horizontal line obtain the values of L and T and
draw a graph with T2 as abscissa and L as ordinate.
The graph would be a straight line. By taking a
convenient point on the graph, g may be calculated.
Similarly, to calculate the value of K, determine the
length AC, BC or CD, CE of the line
ABCDE and compute √12ΧB2 or √24Χ25. Repeat the
procedure for each horizontal line.

19. OBSERVATION AND CALCULATION

20. Computation of proportional
error:
We have from Eq. (10)
Since L = Lx/2 (Lx= AD+BE) and T = t/20,
therefore, we can calculate the maximum
proportional error in the measurement of g as
follows

21. DATA TABLE

22. DATA TABLE

23. Precautions and Discussions:
(i) Ensure that the pendulum oscillates in a vertical
plane and that there is no
rotational motion of the pendulum.
(ii) The amplitude of oscillation should remain
within 40
of arc.
(iii) Use a precision stop-watch and note the time
accurately as far as possible.
(iv) Make sure that there is no air current in the
vicinity of the pendulum.